## Topological Methods in Nonlinear Analysis

### Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$

Vincenzo Ambrosio

#### Abstract

By using variational methods, we investigate the existence of $T$-periodic solutions to \begin{equation*} \begin{cases} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in } (0,T)^{N}, \\ u(x+Te_{i})=u(x) &\mbox{for all } x \in \mathbb{R}^N, \ i=1, \dots, N, \end{cases} \end{equation*} where $s\in (0,1)$, $N> 2s$, $T> 0$, $m\geq 0$ and $f$ is a continuous function, $T$-periodic in the first variable, verifying the Ambrosetti-Rabinowitz condition, with a polynomial growth at rate $p\in (1, ({N+2s})/({N-2s}))$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 75-103.

Dates
First available in Project Euclid: 11 April 2017

https://projecteuclid.org/euclid.tmna/1491876022

Digital Object Identifier
doi:10.12775/TMNA.2016.063

Mathematical Reviews number (MathSciNet)
MR3635638

Zentralblatt MATH identifier
06773117

#### Citation

Ambrosio, Vincenzo. Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 75--103. doi:10.12775/TMNA.2016.063. https://projecteuclid.org/euclid.tmna/1491876022

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